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A multi-Frey approach to some multi-parameter families of Diophantine equations. (English) Zbl 1156.11014

Given a ternary Diophantine equation over \(\mathbb Z\) involving powers of integers, which a priori cannot be solved by classical techniques, it is nowadays useful to try to associate to it a Frey curve, that is an elliptic curve over \(\mathbb Q\) whose arithmetic properties allow to get informations about the resolution of the equation. Of course, it is not at all obvious to find such an elliptic curve, and generally, even if it should be made more precise, it is probably not possible to construct such a curve.
In this paper, the authors solve multi-parametric families of Thue equations, by the simultaneous use of several Frey curves. For instance, they solve Thue equations of the shape \[ 5^ux^n-2^r3^sy^n=\pm 1, \] in non-zero integers \(x\), \(y\) and integers \(u,r,s\geq 1\) and \(n\geq 3\). This equation involves six unknowns.
They prove that the set of solutions, with \(x,y>0\), is given by \[ (u,r,s)\in \Big\{ (1,1,1),(2,3,1)\Big\}, \quad \;n\;\text{arbitrary}\quad \text{and}\quad (x,y)=(1,1). \] Their approach, apart from the modular techniques, uses a recent estimate for linear forms in three logarithms obtained by Mignotte, and a theorem of Bennett which asserts that if \(a, n\) are integers with \(a\neq 0,-1\) and \(n\geq 3\), then the equation \[ \big|(a+1)x^n-ay^n\big|=1 \] has exactly one solution in positive integers \(x,y\), which is given by \(x=y=1\).

MSC:

11D61 Exponential Diophantine equations
11D59 Thue-Mahler equations
11F80 Galois representations
11J86 Linear forms in logarithms; Baker’s method
11Y50 Computer solution of Diophantine equations

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